Is Kähler current class representable by semipositive forms?

A compact complex manifold is called in Fujiki class $$\mathcal C$$ if it is bimeromorphic to a compact Kähler manifold, or equivalently, if there exists a proper holomorphic bimeromorphic map (i.e. a holomorphic modification) $$\mu:\tilde X\to X$$ such that $$\tilde X$$ is a compact Kähler manifold. Another characterization is that if and only if $$X$$ admits a Kähler current, that is a closed (1,1) current $$T$$ satisfying $$T\ge\varepsilon\omega$$ for some real number $$\varepsilon>0$$ and some positive Hermitian form $$\omega$$ (see for example Demailly-Paun 04, p.1263).

As we know the de Rham class $$[T]$$ of the Kähler current $$T$$ is also representable by a smooth form $$\alpha$$, such that $$[\alpha]=[T]\in H^{1,1}(X,\mathbb R)$$, then what property does $$\alpha$$ have? Of course it should not be positive, otherwise the manifold is already Kähler, but except that, what other properties does $$\alpha$$ have, can we always find a semi-positive $$\alpha$$ to represent the class $$[T]$$ of a Kähler current $$T$$?

This is an answer to your last question. In general we can't represent the class of a Kähler current by a semi-positive smooth form $$\alpha$$. Consider the blowup $$\tilde{S} \to S$$ of a compact Kähler surface at a point and let $$E \subset \tilde{S}$$ be the exceptional divisor. Then $$[E] + \varepsilon [\omega] \in H^{1,1}(\tilde{S},\mathbb{R})$$ is the class of a Kähler current for every Kähler form $$\omega$$ on $$\tilde{S}$$ and $$\varepsilon > 0$$. If $$[E] + \varepsilon [\omega]$$ is represented by a semi-positive smooth form $$\alpha$$, then $$-1 + \varepsilon \int_{E}\omega = [E] \cdot ([E] + \varepsilon [\omega]) = \int_{E} \alpha \ge 0,$$ which is impossible for small $$\varepsilon$$.

• In your case, both $S$ and $\tilde S$ are Kähler？But actually what I want to know is that for a non-Kähler manifold in Fujiki class $\mathcal C$, the class of a Kähler current can always be represented by a semipositive form? Sorry for the confusion.
– Tom
May 15, 2022 at 4:38
• You can replace $S$ by any manifold $X$ you like. If $\omega$ is a Kähler current on the blowup $\tilde{X}$, then $[E] + \varepsilon [\omega]$ for $\varepsilon > 0$ is represented by a Kähler current where $E \subset \tilde{X}$ is the exceptional divisor. Again for small $\varepsilon$, we can't represent $[E] + \varepsilon [\omega]$ by a smooth semipositive form because $[\ell] ([E] + \varepsilon [\omega]) < 0$ where $\ell$ is a line in $E$. You can construct non-Kähler $\tilde{X}$ by e.g. blowing up Hironaka's example at a general point.
– HYL
May 15, 2022 at 9:38
• For the holomorphic modification $\mu:\tilde X\to X$, we say $X$ is in Fujiki class $\mathcal C$ if $\tilde X$ is a compact Kähler manifold. Then the Kähler current $T$ should be on $X$ not $\tilde X$, right?
– Tom
May 15, 2022 at 11:39
• If $X$ is in the Fujiki class $\mathcal{C}$, so is $\tilde{X}$. Hironaka's example is Moishezon, in particular in the Fujiki class $\mathcal{C}$.
– HYL
May 16, 2022 at 2:54